Asymptotic Log-harnack Inequality for the 2d Stochastic Cahn-hilliard-navier-stokes System With Degenerate Noise

Abstract

In this article, we consider a Cahn-Hilliard-Navier-Stokes equations in a two dimensional bounded domain and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with the Cahn-Hilliard-Navier-Stokes system driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity. Mathematics Subject Classification. 35R60,35Q35,60H15,76M35,86A05.